Frequency errors and phase noise are present in any satellite communication system. A receiving station in a satellite communication system often receives a transmission with a frequency error, i.e., at a somewhat different frequency than the expected frequency. This frequency error can result from frequency inaccuracies in any of the transmitting station (transmitter), the receiving station (receiver) and the satellite, phase noise introduced by frequency synthesizers at any of the transmitter, the receiver and the satellite, or motion of any of the transmitter, the receiver and the satellite (e.g., due to the Doppler Effect).
For efficient reception, receivers typically use coherent demodulation, which requires estimation of the frequency and phase of the received signal. However, both frequency errors and phase noise vary in time, each at a different rate. If the respective change rates are relatively low while the effective frequency error and phase are estimated, the effective frequency error and phase are often considered as time invariant (i.e., constant). Such an assumption is often used for burst transmissions.
In many systems, phase and frequency offsets are estimated using pilot symbols within the transmission. The pilot symbols are known both to the transmitter, which inserts them into the transmission, and to the receiver, which uses them. Estimation performance improves (i.e., the minimal signal to noise ratio (SNR) needed for an accurate enough estimation becomes lower) as the number of pilot symbols increases. On the other hand, increasing the number of pilot symbols also increases the transmission overhead, thus reducing the efficiency at which the satellite resource is utilized. Hence, minimizing the number of pilot symbols is often desired.
In order to eliminate the increase in transmission overhead due to using pilots, in some systems phase and frequency offsets are estimated based on non-pilot symbols. Extracting phase and frequency information from received non-pilot symbols requires applying a nonlinear operation on the received symbols for removing the modulation information. However, applying such nonlinear operation results also in an SNR reduction (also known as Squaring Loss). This SNR reduction increases as the original SNR becomes lower and as the modulation constellation size increases.
The Cramer Rao lower bound for frequency estimation mean square error (MSE) is given by (Eq. 1):
      E    ⁢          {                        (                                    f              ^                        -                          f              t                                )                2            }        =            3              2        ⁢                  π          2                ⁢                  N                      3            ⁢                                                                      ·          1      SNR        ·                  (                  1                      T            Δ                          )            2      
Where E {x} represents the expectation of x, {circumflex over (f)} represents the estimated frequency, ft represents the true frequency, N represents the number of samples used for deriving the frequency estimation, TΔ represents the sampling interval, and SNR represents the signal to noise ratio.
From (Eq. 1) it is clear that a longer sampling interval TΔ reduces the MSE of the frequency estimation (i.e., brings the estimation closer to the true value). However there is a limit on the length of the sampling interval TΔ, since the frequency offset introduces a phase rotation and that phase rotation after TΔ has to be smaller than it in order to avoid ambiguity in the frequency estimation. Thus if pilot symbols are used, the choice of spacing between pilot symbols is a tradeoff between supporting a large frequency offset (for which the interval between pilot symbols has to be short in order to avoid ambiguity) and obtaining high estimation accuracy (for which the interval between pilot symbols has to be longer).
Implementation complexity plays an important role in satellite communication systems. Low computational complexity solutions are considered advantageous and can have direct influence on hardware cost. Hence low computational complexity phase and frequency estimators are often desired.
Maximum likelihood frequency offset estimation involves rotating received pilot symbols and/or the results of a nonlinear operation on unknown symbols according to all frequency offset hypotheses, summing the rotated symbols, and then choosing the hypothesis that attains the maximum absolute value. The frequency estimation resolution is dictated by the minimal difference between frequency hypotheses. As frequency estimation resolution becomes higher (i.e., the difference in frequency units between adjacent frequency offset hypotheses (Δ) becomes smaller) and as the frequency offset range (Ω) becomes higher, more hypotheses have to be tested. As the computational complexity increases with the number of hypotheses tested, the computational complexity is proportionate to the frequency offset range and to the required frequency estimation resolution (e.g., to the ratio Ω/Δ).
Unfortunately, the frequency offset range and the desired estimation resolution are often both high, thus the required number of hypotheses is extremely large and the frequency estimator is impractical for implementation. Thus, a low complexity frequency (offset) estimator that attains accurate estimation with a minimal amount of pilot symbols and supports a large frequency offset range is desired.